bigint-gcd

Greater common divisor (gcd) of two BigInt values using Lehmer's GCD algorithm. See https://en.wikipedia.org/wiki/Greatest_common_divisor#Lehmer's_GCD_algorithm. On my tests it is faster than Euclidean algorithm starting from 80-bit integers.

A version 1.0.2 also has something similar to "Subquadratic GCD" (see https://gmplib.org/manual/Subquadratic-GCD ), which is faster for large bigints (> 65000 bits), it should has better time complexity in case the multiplication is subquadratic, which is true in Chrome 93.

Installation

$ npm install bigint-gcd

Usage

import gcd from './node_modules/bigint-gcd/gcd.js';

console.log(gcd(120n, 18n));

There is also an implementation of the Extended Euclidean algorithm, which is useful to find the multiplicative modular inverse:

console.log(gcd.gcdext(3n, 5n)); // [2n, -1n, 1n]

And "Half GCD" which is useful to do the Rational reconstruction: It returns the transformation matrix and the transformed values after applying about half of the Euclidean steps.

console.log(gcd.halfgcd(1000000n, 1234567n)); // [-16n, 13n, 21n, -17n, 49371n, 12361n]

Performance:

The benchmark (see benchmark.html) resutls under Chrome 131:

| bit size | gcd | gmpy2 gcd | invmod | gmpy2 invert | | ------------------ | ------------------- | ------------------ | ------------------ | ------------------ | | 64 | 0.000270ms | 0.00030ms | 0.000310ms | 0.00066ms | | 128 | 0.001270ms | 0.00047ms | 0.001720ms | 0.00137ms | | 256 | 0.002660ms | 0.00153ms | 0.003650ms | 0.00224ms | | 512 | 0.005460ms | 0.00321ms | 0.007630ms | 0.00391ms | | 1024 | 0.012080ms | 0.00653ms | 0.018250ms | 0.00806ms | | 2048 | 0.031130ms | 0.01429ms | 0.048220ms | 0.01587ms | | 4096 | 0.067870ms | 0.02979ms | 0.137700ms | 0.03590ms | | 8192 | 0.174320ms | 0.06837ms | 0.341310ms | 0.09035ms | | 16384 | 0.503910ms | 0.17093ms | 0.867190ms | 0.24908ms | | 32768 | 1.677730ms | 0.49816ms | 2.281250ms | 0.75801ms | | 65536 | 4.406250ms | 1.43795ms | 6.152340ms | 1.94962ms | | 131072 | 11.828130ms | 3.98527ms | 16.937500ms | 4.98559ms | | 262144 | 32.296880ms | 10.52619ms | 47.203130ms | 14.05025ms | | 524288 | 86.625000ms | 28.16362ms | 123.500000ms | 38.94622ms | | 1048576 | 213.312500ms | 70.89262ms | 310.062500ms | 103.71075ms | | 2097152 | 519.250000ms | 177.16650ms | 773.875000ms | 269.43650ms | | 4194304 | 1255.750000ms | 433.85675ms | 1870.500000ms | 658.39875ms | | 8388608 | 2988.500000ms | 1069.74050ms | 4548.000000ms | 1673.88250ms |

Benchmark:

import {default as LehmersGCD} from './gcd.js';

function EuclideanGCD(a, b) {
  while (b !== 0n) {
    const r = a % b;
    a = b;
    b = r;
  }
  return a;
}

function ctz4(n) {
  return 31 - Math.clz32(n & -n);
}
const BigIntCache = new Array(32).fill(0n).map((x, i) => BigInt(i));
function ctz1(bigint) {
  return BigIntCache[ctz4(Number(BigInt.asUintN(32, bigint)))];
}
function BinaryGCD(a, b) {
  if (a === 0n) {
    return b;
  }
  if (b === 0n) {
    return a;
  }
  const k = ctz1(a | b);
  a >>= k;
  b >>= k;
  while (b !== 0n) {
    b >>= ctz1(b);
    if (a > b) {
      const t = b;
      b = a;
      a = t;
    }
    b -= a;
  }
  return k === 0n ? a : a << k;
}

function FibonacciNumber(n) {
  console.assert(n > 0);
  var a = 0n;
  var b = 1n;
  for (var i = 1; i < n; i += 1) {
    var c = a + b;
    a = b;
    b = c;
  }
  return b;
}

function RandomBigInt(size) {
  if (size <= 32) {
    return BigInt(Math.floor(Math.random() * 2**size));
  }
  const q = Math.floor(size / 2);
  return (RandomBigInt(size - q) << BigInt(q)) | RandomBigInt(q);
}

function test(a, b, f) {
  const g = EuclideanGCD(a, b);
  const count = 100000;
  console.time();
  for (let i = 0; i < count; i++) {
    const I = BigInt(i);
    if (f(a * I, b * I) !== g * I) {
      throw new Error();
    }
  }
  console.timeEnd();
}

const a1 = RandomBigInt(128);
const b1 = RandomBigInt(128);

test(a1, b1, LehmersGCD);
// default: 426.200927734375 ms
test(a1, b1, EuclideanGCD);
// default: 1136.77294921875 ms
test(a1, b1, BinaryGCD);
// default: 1456.793212890625 ms

const a = FibonacciNumber(186n);
const b = FibonacciNumber(186n - 1n);

test(a, b, LehmersGCD);
// default: 459.796875 ms
test(a, b, EuclideanGCD);
// default: 2565.871826171875 ms
test(a, b, BinaryGCD);
// default: 1478.333984375 ms